Linear Analysis for Determining and Visualizing Critical Thermal Boundaries of Power Systems Robert Entriken 1 1 2 Walter Murray Tomas Tinoco De Rubira Electric Power Research Institute 2 2 Stanford University Introduction Recommended Power Adjustments Current security assessment practices typically require computationally expensive system simulations rely significantly on operator experience Define measure of system security X T M (y) := log(qi y − ti) (N are nearest boundaries) i∈N ∇M (y) gives power adjustment direction for improving security may not be practical, can use sparse approximations However, systems are becoming more stressed and unpredictable, hence require faster and more frequent security analyses require considering and monitoring more potentially critical quantities This work explores fast techniques based on linear system model for identifying critical thermal (and voltage) boundaries visualization techniques for enhancing operator awareness of system security 1 Model Accuracy Model accuracy near boundaries tested on two 2.5k-bus systems 2 Modeling Approach The system model considered is given by Ax = b + Y y, Cx ≥ d, y min ≤y≤y max x is system state, y is vector of generator and load powers equalities are linearized power flow equations inequalities are voltage and linearized thermal bounds (based on current magnitudes) Updating model with information obtained from linearizing system at new locations helps reduce errors Visualization Can be expressed in terms of y only Qy ≥ t, y min ≤y≤y max Strategy determine a suitable plane P reconstruct nearest boundaries on that plane 1 allows expressing boundary distances in terms of power changes Q is dense, so not constructed! 2 Nearest Boundaries Identifying nearest boundaries requires comparing distances from current point y0 (generator and load powers) requires knowing kqk2 for each row q of Q T 2 exploit VAR(q z) = kqk2, where z are i.i.d. N (0, 1) estimate kqk2 through sampling and matrix-vector products with Q What plane should be used? One that tries to preserve boundary distances, giving more emphasis to nearest ones, e.g., “security plane” P := span{u1, u2}, u1 and u2 are left singular vectors of Ms := [w1q1 · · · wmqm] , qT y = t wi are weights inversely proportional to boundary distance. q T q y 0− t ‖q‖2 y0 Experiment: identify 500 nearest boundaries of 45k-bus system (117k thermal boundaries, 90k voltage boundaries) algorithm time (s) samples % top 500 % top 50 sampling 1.28 ± 0.09 53 ± 4 95 ± 2 100 ± 0 full Q 4836.34 100 100 Conclusions Proposed techniques are efficient and effective for identifying and visualizing critical thermal boundaries Linear system model is accurate for many but not all of the critical thermal boundaries ISGT Conference February 18, 2015 Washington, DC